Question

Find the quantifier which best describes the variable of the open sentence ${x^2} + 2 \geqslant 0$ .
A.Universal
B.Existential
C.Neither (A) nor (B)
D.Does not exist

Answer 1

Hint: First write the given open statement using the phrase “for every” and “there exists”.
After that, check which of the statements suits well to the given open statement.
Thus, decide whether universal quantifier or existential quantifier describes the open statement ${x^2} + 2 \geqslant 0$ .

Complete step-by-step answer:
The open source sentence given here is ${x^2} + 2 \geqslant 0$ .
So, first we will write the open source statement as “For every real number x, ${x^2} + 2 \geqslant 0$ .” and “There exists a real number x such that ${x^2} + 2 \geqslant 0$ .”
Now, we will check which one of the above two statements suits well to the given open statement. And decide the correct quantifier accordingly.
Here, the statement “For every real number x, ${x^2} + 2 \geqslant 0$ .” suits well to the given open statement.
Thus, the universal quantifier best describes the variable of the open statement.
So, option (A) Universal is correct.

Note: Universal quantifier:
The phrase “for every” (or its equivalents) is called universal quantifier. The symbol $\forall $ is used to denote a universal quantifier. For example, $\left( {\forall x,P\left( x \right)} \right)$ can be stated as “For every x, \[P\left( x \right)\] ”, where \[P\left( x \right)\] is a predicate. Thus, for every value of x in the universal set, \[P\left( x \right)\] is true.
Existential quantifier:
The phrase “there exists'' (or its equivalents) is called existential quantifier. The symbol $\exists $ is used to denote an existential quantifier. For example, $\left( {\exists x,P\left( x \right)} \right)$ can be stated as “There exists and x such that \[P\left( x \right)\] ”, where \[P\left( x \right)\] is a predicate. Thus, there is at least one value of x in the universal set for which \[P\left( x \right)\] is true.